MODULAR FORMS (PRELIMINARY VERSION)
PAYMAN L KASSAEI
In these notes, we explain some recent progress on analytic continuation of overconvergent p-adic Hilbert modular forms and applications. We will begin with the classical case of elliptic modular forms to explain the basic ideas and hint at what new ideas are needed in the general case. We then move on to the case of Hilbert modular forms where the prime p is unramified in the relevant totally real field.
1. The classical case
1.1. In [BT99], Buzzard and Taylor proved the modularity of a certain kind of a Galois rep- resentation ρ by showing first that ρ arises from an overconvergent modular form f, and then proving that f is indeed a classical modular form. The proofs of classiality in this work and the subsequent generalization by Buzzard [Buz03] were through analytic continuation of f from its original domain of definition (which is an admissible open region in the rigid analytic modular curve) to the entire modular curve. This implies classicality since by the rigid analytic GAGA, any global analytic section of a line bundle over the analytification of a projective variety is, indeed, algebraic.
Earlier, in [Col96], Coleman had proved a criterion for classicality of p-adic overconvergent modular forms in terms of slope, i.e., the p-adic valuation of the eigenvalue of the Up Hecke operator.
Theorem 1.1.1. (Coleman) Any overconvergent modular formf of weight kand slope less than k−1 is classical.
Coleman’s proof involved calculations with the cohomology of modular curves. We could, however, ask whether this result could be proven by invoking the above principle of analytic continuation. In other words, given the slope condition, could we analytically continuef from its domain of definition to the entire modular curve? In [Kas06], we showed that this is possible and involves the construction of a series whose convergence is guaranteed by the given slope condition.
In this section, we will explain the proof by dissecting the method to see what is essential for the application of the method in more general cases. In doing so, we will introduce some ideas of Pilloni which allows for a less explicit and, hence, more general approach.
1.2. The proof of Coleman’s theorem via analytic continuation [Kas06]. In this section only, we letY denote the completed modular curve of level Γ1(N)∩Γ0(p) defined overQp, where N ≥4 is an integer prime top. Its noncuspidal locus classifies the data (E, H) overQp-schemes, where E is an elliptic curve with Γ1(N)-level structure, and H a finite flat subgroup scheme of E of order p. Let ω be the usual sheaf on Y whose sections are invariant differentials on the
1
universal family of (generalized) elliptic curves onY. Modular forms of level Γ1(N)∩Γ0(p), and weight k∈Zare elements of H0(Y, ωk). We let Yan denote the p-adic rigid analytification ofY, and continue to denote the analytification of ω by ω.
Let Yan,0 denote the modular curve whose noncuspidal locus classifies all (E, H, D) such that (E, H)6= (E, D) and both are classified by Yan. There are two morphisms π1, π2 :Yan,0 →Yan sending (E, H, D) to (E, H) and (E/D,H), respectively, where ¯¯ H denotes the image of H in E/D.
To define regions insideYan, we need to recall the notion of degree of a finite flat group scheme over a finite extension of Qp and some of its properties.
The degree of a finite flat group scheme. We define the notion of degree and record some properties that we will use later. This useful notion was defined by Illusie and others, and has been more recently studied by Fargues in [Far10].
Definition 1.2.1. LetOK be the ring of integers in a finite extensionK ofQp. Let Λ be a finite torsion OK-module. Choose an isomorphism Λ ∼= ⊕di=1OK/(ai) with ai ∈ OK. We define the degree of Λ to be deg(Λ) = Pd
i=1νp(ai). This definition can be shown to be independent of the choice of the above isomorphism.
If G is finite flat group scheme over OK, we define deg(G) = deg(ωG), where ωG is the OK- module of global invariant differentials on G. It can be shown [Far10, §3] that deg(G) equals the p-adic valuation of a generatorδG of F itt0(ωG), the zeroth fitting ideal ofωG.
We record some lemmas which we will use later.
Lemma 1.2.2. [Far10, Lemme 4] Assume that 0→G0 →G→G00→0 is an exact sequence of finite flat group schemes over OK. We have deg(G) = deg(G0) + deg(G00).
Lemma 1.2.3. [Far10] Let λ:A→B be an isogeny ofp-power degree between abelian schemes over S = Spec(OK). LetG be the kernel of λ. LetωA/S and ωB/S denote the conormal sheaves of A andB, respectively. Then
deg(G) =νp(det(λ∗ :ωB/S →ωA/S)).
In particular, if A is an abelian scheme over Spec(OK) of dimension g, then deg(A[pn]) =ng.
Proposition 1.2.4. [Far10, Corrolaire 3]Let GandG0 be two finite flat group schemes over S= Spec(OK), and λ:G→G0 a morphism of group schemes which is generically an isomorphism.
Then, deg(G)≤deg(G0) and the equality happens if and only ifλ is an isomorphism.
Proposition 1.2.5. [Pil11, Lemme 2.3.4] If G is a truncated Barsotti-Tate group of level 1 defined over a finite extension of Qp, then deg(G) is an integer.
The degree function can be used to parameterize points on the modular curve, and to cut out rigid analytic subdomains on it.
Definition 1.2.6. If Q = (E, H) is point on Yan, we define deg(Q) = deg(H), if Q has good reduction. Otherwise, we define deg(Q) = 0 or 1, depending on whether Q has ´etale or mul- tiplicative reduction. If I is a subinterval of [0,1], we define YanI to be the admissible open subdomain ofYanconsisiting of pointsQsuch that deg(Q)∈I. Ifa, bare rational numbers, then
Yan[a, b] is quasi-compact. It is easy to see that the locus of supersingular location is exactly Yan(0,1). The ordinary locus has two connected components, the multiplicative locus Yan[1,1], and the ´etale locus, Yan[0,0]. An overconvergent modular form of weight k ∈ Z is a section of ωk on Yan[1−,1] for some >0.
Remark 1.2.7. There is a simple relationship between the degree function and the function v0 defined by Buzzard in [Buz03, §4]. We havev0(E, H) = 1−deg(E, H).
Given the above lemma, we can now rephrase the classical theory of canonical subgroups (due to Katz and Lubin) in terms of degrees, as follows:
Proposition 1.2.8. (Lubin-Katz) Let Q = (E, H) ∈ Yan. Define Sib(Q) = {Q0 = (E, H0) ∈ Yan :Q0 6=Q}.
• If deg(Q) >1/(p+ 1), then for any Q0 ∈Sib(Q), we have deg(Q0) = (1−deg(Q))/p <
1/(p+ 1).
• If deg(Q) = 1/(p+ 1), then for any Q0 ∈Sib(Q), we have deg(Q0) = 1/(p+ 1).
• Ifdeg(Q)<1/(p+ 1), then there is a unique(E, H0) =Q0 ∈Sib(Q), such thatdeg(Q0)>
1/(p+ 1); H0 is called the (first) canonical subgroup of E, it varies analytically with respect to Q, and we have deg(Q0) = 1−pdeg(Q). For all other Q00 ∈Sib(Q), we have deg(Q00) = deg(Q)<1/(p+ 1).
We make a definition:
Definition 1.2.9. If deg(E, H) < pm−11(p+1), then, for any 1 ≤ n ≤ m, we can define a cyclic subgroupCnofE[pn] of order pn, called then-th canonical subgroup ofE, inductively as follows.
By Proposition 1.2.8,E has a first canonical subgroupC1, and deg(E/C1,H) = 1¯ −deg(E, C1) = pdeg(E, H)< pm−21(p+1). Hence, by induction, we can constructCn0, then-th canonical subgroup ofE/C1, for all 1≤n≤m−1. For 2≤n≤m, we defineCn= pr−1(Cn−10 ), where pr :E →E/C1 is the projection.
The first step of the analytic continuation– the first take. This step is due to Buzzard [Buz03]. Using an iteration of the Up operator, Buzzard extends f from its initial domain of definition to progressively larger domains, eventually extending f toYan(0,1].
Proposition 1.2.10. (Buzzard) Let f be an overconvergent modular form f satisfying Up(f) = apf withap 6= 0. Then f extends analytically toYan(0,1].
We first recall the definition of the Up operator. Let V1 and V2 be admissible opens of Yan such that π1−1(V)⊂π−12 (W) inside Yan,0. We define an operator
Up =UWV :ωk(W)→ωk(V), via the formula
(1.2.1) Up(f) = 1
pπ1,∗(res(pr∗π∗2(f))),
where resis restriction from π2−1(W) to π1−1(V), π1,∗ is the trace map associated with the finite flat map π1, and pr∗ : π2∗ωk→π1∗ωk is a morphism of sheaves on Yan, which at (E, H, D) is induced by pr∗ : ΩA/D →ΩAcoming from the natural projection pr :A→A/D.
One can also define a set-theoretic Up correspondence as the map which sends a subset S ⊂ Yan to another subset Up(S) = π2(π1−1(S)). The condition π1−1(V) ⊂ π−12 (W) is equivalent to Up(V)⊂ W.
The principle underlying Buzzard’s method is the following. Let W be an admissible open such that Up(W) ⊆ W. Suppose f is defined over W and Up(f) = apf with ap 6= 0. Suppose further that V ⊃ W is an admissible open subset ofYan such thatUp(V)⊆ W. Then,f extends from W to V, and the extended section (which we continue to denote by f) satisfies the same functional equationUp(f) =apf. The reason for this is simple: the extension off shall be taken to be a1
pUp(f) and can be checked to satisfy all the desired properties. Therefore, a strategy for extending f to an admissible open U ⊇ W is to prove that successive application of Up sends U into W. For details of this construction in general, see [Kas09,§3.1].
Proof of Proposition 1.2.10. We invoke the above principle and draw all the shrinkage under Up we need from the degree calculations in Proposition 1.2.8.
If α is such that 1/(p+ 1)<1−α <1, then Proposition 1.2.8 shows that Up(Yan[1−α,1])⊂Yan[1−α
p,1].
This implies that forM large enough,UpM sendsYan(1−α,1] inside a domain of definition off, and, hence,f extends to Yan(1−α,1]. Repeating this argument for all suchα, we can extend f to a section (denoted f again) on Yan(p+11 ,1] satisfyingUp(f) =apf. By Proposition 1.2.8
Up(Yan[ 1
p+ 1,1])⊂Yan( 1 p+ 1,1].
Hence, we deduce thatf extends further to Yan[p+11 ,1], satisfying stillUp(f) =apf. Finally, for any 0< β <1/(p+ 1), Proposition 1.2.8 implies that
Up(Yan[β,1])⊂Yan[pβ,1]).
Applying this successively, we deduce that a high enough power of Up will send Yan[β,1] inside Yan[p+11 ,1], and, hence, f can be extended to Yan[β,1]. Applying this to all β > 0, we get the desired result.
The first step of the analytic continuation– the 2nd take. Proposition 1.2.8 allows a precise calculation of the Up correspondence in terms of the degree parametrization on the modular curve. This calculation was used in the above proof. In more general situations such calculations could prove difficult to carry out. In this section we explain, `a la Pilloni, how Buzzard’s proof given above does not really need the full force of the degree calculations under the Up correspondence.
Looking at the above proof, we can readily see that the correspondenceUp increases degree in the cases considered. This is in fact a general principle.
Proposition 1.2.11. Let Q∈Yan, and Q0 ∈Up({Q}). Then deg(Q0)≥deg(Q).
Proof. This is an immediate consequence of Proposition 1.2.4. The morphism H→H¯ induced by A→A/D is generically an isomorphism asH∩D={0} generically.
Looking more closely at the proof in the last section, we have in fact shown that Up increases the degreestrictly on the non-ordinary locus ofYan. This is possible to prove in light of explicit calculations afforded by Proposition 1.2.8. In fact, this is exactly what makes the proof work: by iterating Up enough times, any point in the non-ordinary locus will eventually land close enough to Yan[1,1] where f will be defined. The following approach, due to Pilloni, achieves the same without appealing to Proposition 1.2.8. This approach is useful in cases where an analogue of Proposition 1.2.8 is not readily available.
Proposition 1.2.12. LetQ= (E, H)∈Yan defined overOK. If there isQ0∈Up({Q})such that deg(Q) = deg(Q0), then H is a truncated Barsotti-Tate group of level 1. In particular,deg(Q) is an integer (by Proposition 1.2.5).
Proof. LetQ0 = (E/D,H), and assume w.l.o.g that¯ Q0 is also defined overOK. SinceH →H¯ is generically an isomorphism, we have deg(Q0) = deg( ¯H)≥deg(H) = deg(Q). Since the equality happens, by Proposition 1.2.4, we know that H→H¯ must be an isomorphism over OK. This implies thatE[p]∼=H×D, and hence both Hand Dare truncated Barsotti-Tate groups of level
1.
Corollary 1.2.13. In the situation of Proposition 1.2.12, we havedeg(Q)∈ {0,1}, and henceQ belongs to the ordinary locus. In other words, over the non-ordinary locus of Yan, Up increases degrees strictly.
Proof. By Proposition 1.2.12, we must have deg(Q)∈Z. Since deg(Q)∈[0,1], the claim follows.
This gives another proof of the fact that Up increases degree strictly over the non-ordinary locus ofYan. We can now present a second proof of Proposition 1.2.10, due to Pilloni.
Second proof of Proposition 1.2.10. Assume f is defined on Yan[1−,1], for some rational > 0. It is enough to show that for any rational α ∈ (0,1−), there is r ∈ N such that Upr(Yan[α,1])⊂Yan[1−,1]. This follows immediately if we show that there is a positive tsuch that Up increases degree bytover the entireYan[α,1−].
Let pr :A → A/H be the universal isogeny over Yan. Let ωA, ωA/H denote, respectively, the determinants of the conormal sheaves of A,A/H over Yan. Set L = ω−1A/H⊗ωA, which is an invertible sheaf on Yan. The morphism pr∗ : ωA/H→ωA defines a section δ of L on Yan. By Lemma 1.2.3, for anyQ∈Yan, we have deg(Q) =νp(δ(Q)).
Consider now the sectionδ0 =π1∗δ⊗(π2∗δ)−1 ∈H0(Yan,0, π1∗L−1⊗π2∗L). By Corollary 1.2.13, we have νp(δ0) >0 over the entire non-ordinary locus. For any rational number α ∈(0,1−), Yan[α,1−] is a quasi-compact rigid analytic domain ofYan, and, hence, π−11 (Yan[α,1−]) is a quasi-compact rigid analytic domain in Yan,0. Therefore, by the maximum modulus principle, νp(δ0) attains a a minimum t over it. This minimum t must be positive as Yan[α,1−] lies entirely inside the non-ordinary locus.
The second step of the analytic continuation. So far, we have seen that as long asν(ap) is finite, we can extendf toYan(0,1]. We now assume thatν(ap)< k−1, and prove the classicality
off. What is left to show is that under this assumptionf can be further extended fromYan(0,1]
to Yan =Yan[0,1]. The missing locus isYan[0,0], i.e., the ordinary ´etale locus. We will do this by constructing a section onF onYan[0,0] and showing that it glues tof onYan(0,1] producing a global section.
To motivate the construction of the extension off toYan, we assume for now thatf is classical of slope less than k−1. Assume (E, H) is inYan[0,0]. SinceUpf =apf, we can write
f(E, H) = 1 pap
X
H∩D1=0
pr∗f(E/D1,H),¯ (1.2.2)
where the sum is over the cyclic subgroups H1 of rank p which intersect H trivially, and ¯H denotes the image ofH inE/D1. SinceH is not canonical, all but one of points appearing on the right hand side of the above formula belong to Yan[1,1] (by Proposition 1.2.8). The exceptional term corresponds to D1=C1, the first canonical subgroup ofE. Applying the above formula to (E/C1,H) we get¯
f(E/C1,H) =¯ 1 pap
X
H∩D2=0,C1⊂D2
pr∗f(E/D2,H),¯ (1.2.3)
where the sum is over the cyclic subgroups D2 of rank p2 which contain C1 and intersect H trivially. We, hence, find
f(E, H) = 1 pap
X
H∩D1=0,D16=C1
pr∗f(E/D1,H) + (¯ 1 pap
)2 X
H∩D2=0,C1⊂D2
pr∗f(E/D2,H).¯
Similarly, we find that the only point appearing in this expression that doesn’t belong to Yan[1,1] is (E/C2,H). We will repeat this process with¯ f(E/C2,H), and keep going in the same¯ way. At the n-th step, we separate the term corresponding to the quotient ofE byCn (then-th canonical subgroup of E) from the rest of the terms, and rewrite the term via the functional equation Upf =apf as above. The result is the following.
Proposition 1.2.14. Let f be a classical modular form of level Γ0(N)∩Γ1(p), weight k and slope less than k−1. We have
f(E, H) =
∞
X
n=1
( 1
pap)n X
Dn
pr∗f(E/Dn,H)¯ , (1.2.4)
where Dnruns through all the cyclic subgroups ofE of rankpn which containCn−1, are different fromCn, and intersectH trivialy. All the points appearing in the above series belong to Yan[1,1].
Proof. The only thing left to show is that the series converges. By Lemma 1.2.2, deg(Cn) =n, and, hence, Lemma 1.2.3 implies that pr∗(η) is divisible bypnk for any sectionηofωk. Hence the
“error term” (pa1
p)npr∗f(E/Cn,H) is divisible by (¯ pa1
p)npnk = (pk−1/ap)n which tends to zero as
n goes to infinity by the assumption onap.
Before we proceed, we would like to make a definition to formalize the above “error term” as a special term among the terms appearing in the definition of the Up operator.
Definition 1.2.15. For any interval I ⊂ [0,1/(p+ 1)), and any r > 0, we define Ir to be the interval defined by multiplying all the elements in I byr. Using Proposition 1.2.8, we define
Usp :YanI1p →YanI,
via Usp(E, H) = (E/C1,H), where¯ C1 is the canonical subgroup of E. It induces a morphism Usp :ωk(YanI)→ωk(YanI1p)
defined as Uspf(E, H) = 1p pr∗f(E/C1,H). It follows that for any¯ n∈N, the map (Usp)n:ωk(YanI)→ωk(YanIpn1 )
is given by (Usp)nf(E, H) = (1p)npr∗f(E/Cn,H), where¯ Cn is then-th canonical subgroup ofE.
Assume W,V are admissible opens of Yan satisfying π2−1(W)⊂π1−1(V) so that we have aUp operator UWV : ωk(W)→ωk(V). Whenever there is a decomposition W = W1 ∪ W2, we get a decomposition of the UWV operator to a the sum of two operators UWV
1 : ωk(W1)→ωk(V) and UWV
2 :ωk(W1)→ωk(V). In particular, by virtue of Proposition 1.2.8, we have π−11 (Yan[0,0]) =π2−1(Yan[1,1])∪π2−1(Yan[0,0]),
(1.2.5)
and, correspondingly, the operatorUp :ωk(Yan[1,1])∪ωk(Yan[0,0])→ωk(Yan[0,0]) decomposes as a sum of two operators denoted as follows:
Up =Upnsp+Upsp.
Unravelling the above construction shows thatUpnsp :ωk(Yan[1,1])→ωk(Yan[0,0]) is given by Upnspf(E, H) = 1
p X
D6=C1,H
pr∗f(E/D,H),¯ (1.2.6)
and Upnsp : ωk(Yan[0,0])→ωk(Yan[0,0]) is the map defined in Definition 1.2.15 for I = [0,0].
Using this notation, the discussion above can be summarized as follows: iff is classical, of weight k and of slope less thank−1, then
f|Yan[0,0]=
∞
X
n=1
( 1
ap)n(Upsp)n−1Upnsp(f|Yan[1,1]).
Proof of Theorem 1.1.1. Define F on Yan[0,0] exactly as above.
F =
∞
X
n=1
(1 ap
)n(Upsp)n−1Upnsp(f|Yan[1,1]).
(1.2.7)
The convergence of the series under the slope assumption follows from the same argument as in Proposition 1.2.14. We want to show that F on Yan[0,0] can be glued to f on Yan(0,1].
The problem is that the series defining F can not be extended outsideYan[0,0], as its definition depends on the existence of allCn’s which requires the ordinarity ofE. However, as we shall see,
the partial sums in the series overconverge outside Yan[0,0] (albeit to extents that vanish in the limit), and it gives us enough information to prove the gluing.
In fact, for any 0≤t <1/(p+ 1), Proposition 1.2.8 gives a decomposition similar to 1.2.5:
π1−1(Yan[0,t
p]) =π2−1(Yan[1− t
p,1])∪π−12 (Yan[0, t]).
(1.2.8)
Therefore, the above decomposition of Up extends from Yan[0,0] to Yan[0,pt], and we can write Up=Upsp+Upnsp, where
Upnsp :ωk(Yan[1− t
p,1])→ωk(Yan[0,t p]), is given by the same formulae as 1.2.6, and
Upsp:ωk(Yan[0, t])→ωk(Yan[0, t p])
is as in Definition 1.2.15. Let us fix a rational number 0< t0 <1/(p+ 1), and define V :=Yan(0, t0]
Also, for any m≥0, define
Sm† :=Yan[0, t0 pm].
The above decomposition of Up allows us to define a section ofωk on Sm†, form≥1, as follows Fm =
m
X
n=1
( 1
ap)n(Upsp)n−1Upnsp(f|
Yan[1−t0 p ,1]
).
In other words, Fm is the m-th partial sum of F which overconverges from Yan[0,0] to Sm†. We want to show that these partial series become very close to f outside Yan[0,0]. An easy calculation shows us that for m≥1, we have the following equality onV ∩ Sm† =Yan(0,ptm0 ]:
f =Fm+ (1
ap)m(Upsp)mf|V, (1.2.9)
where Upsp is as in Definition 1.2.15 forI = (0, t0]. We, therefore, need to estimate (Upsp)mf|V. Lemma 1.2.16. The following estimates hold:
• The collection of sections{f|V, Fm :m∈N} is uniformly bounded.
• |f −Fm|onSm† ∩ V=Yan(0,ptm0 ] tends to zero as m goes to infinity.
• |Fm+1−Fm|onSm+1† =Yan[0,pm+1t0 ]tends to zero as m goes to infinity.
Proof. LetZ ⊂Yan[0,p(p+1)1 ), and let h∈ωk(Z). For anyQ= (E, H)∈(Upsp)−1(Z), we denote the first canonical subgroup of E by C1. Letd:= inf{deg(C1) :Q∈(Upsp)−1(Z)}. We can write
|Upsph(Q)| = |1
p pr∗h(Upsp(Q))|
= p−k(deg(C1))|1
p h(Upsp(Q))|
≤ p1−kd|h|Z,
where, in the second equality, we have used Lemma 1.2.3. This implies that
|Upsph|(Usp
p )−1(Z)≤p1−kd|h|Z. (1.2.10)
We first show that |f|V is bounded. For m ≥ 0, let Zm := Sm† − Sm+1† = Yan(pm+1t0 ,ptm0 ].
Proposition 1.2.8 implies that Up(Z0) lies inside (the quasi-compact)Yan[p+11 ,1], over whichf is bounded. This, in turn, implies that f = a1
pUpf is bounded onZ0. On the other hand, we have seen thatf−a1
pUpspf =F1 extends to the quasi-compactS1†, and, hence, it has a bounded norm.
Let M1 denote a common bound for f on Z0 and f −a1
pUpspf on S1†. We prove, by induction, thatf is bounded byM1p
kt0 p +···+kt0
pm onZm. Assume this is true form−1, withm≥1. We have (Upsp)−1(Zm−1) =Zm, and the infimumdintroduced above equals 1−ptm0 on Zm, by Proposition 1.2.8. Hence, inequality 1.2.10 gives us
|1
ap Upspf|Zm ≤pνp(ap)+1−k(1−
t0
pm)|f|Zm−1 ≤p
kt0
pm|f|Zm−1 ≤M1p
kt0 p +···+kt0
pm.
Therefore, |f|Zm ≤max{|f− a1
pUpspf|Zm,|a1
pUpspf|Zm} ≤M1p
kt0 p +···+kt0
pm, as claimed. Now, since V =S
m≥0Zm, it follows that
|f|V ≤M :=M1p
kt0 p−1.
For the second part of the lemma, we apply Equation 1.2.9 along with inequality 1.2.10 with Z =Sm−1† ∩ V to deduce that
|(Upsp)mf|V|S†
m∩V ≤p1−k(1−
t0 pm−1)
|(Upsp)m−1f|V|S† m−1∩V. Induction on m gives us
|( 1 ap
)m(Upsp)mf|V|Sm ≤ | 1 ap
|m pm−k(m−
pt0 p−1)
=pm(νp(ap)−(k−1))p
kpt0
p−1 →0 as m→ ∞ since νp(ap) < k−1. The third statement of the Lemma can be proven in exactly the same way, as Fm+1−Fm = (a1
p)m+1(Upsp)m(F1) from the definition. Finally, the uniform boundedness of the collection {Fm} follows immediately from the above results, along with the fact that the sequence Fm is convergent onYan[0,0].
Finally, we can use the gluing lemma in [Kas06] to show thatf andF glue together to produce a section of ωk over Yan[0, t0]. While the domains of definitions of F and f do not overlap, we can use the above overconvergence results to prove the gluing.
It is enough to show thatf extends fromV =Yan(0, t0] toS0†=Yan[0, t0]. Let ˇObe the sheaf of rigid analytic functions on Yan with norm at most 1. By Lemma 1.2.16, we can rescale (and restrict to a trivializing open cover for ω) to assume that all the Fm’s andf|V are sections of ˇO.
Furthermore, modulo choosing a subsequence, we can assume that
|Fm−f|S†
m∩V ≤(1 p)m.
This implies thatFm and f|V glue modpm to give a sectionhm of ˇO/pmOˇ overSm† ∪ V =S0†. A theorem of Bartenwerfer [Bar70] states that for a smooth quasi-compact rigid analytic varietyZ, we have cH1(Z,OˇZ) = 0 for some scalarc with|c| ≤1. A standard argument shows then that
chm ∈O(Sˇ 0†)/pmO(Sˇ 0†)⊂( ˇO/pmO)(Sˇ 0†).
The compatibility of the chm’s implies that their inverse limit provides an element h of lim←−
m
chm ∈lim←−
m
O(Sˇ 0†)/pmO(Sˇ 0†) = ˇO(S0†).
We define hto be 1/ctimes this section. It is immediate from Lemma 1.2.16 that h|V =f|V and h|Yan[0,0] =F. This ends the proof of Theorem 1.1.1.
1.3. Discussion: the essential ingredients in the second step of analytic continuation.
Since we are interested in applying the above method in more general situations, we would like to discuss some of the ingredients that made the above proof work, in a less case-specific fashion.
We first focus on the construction of the series on Yan[0,0].
As we have seen earlier, the idea of the first step of the analytic continuation does not work on Yan[0,0] as Up does not increase degrees strictly on this domain. In fact, another way to characterize Yan[0,0] is the following:
Yan[0,0] ={(E, H)∈Yan[0,1) :∃!(E/G1,H)¯ ∈Up{(E, H)} s.t. deg(E/G1,H) = deg(E, H¯ )}.
Again, having Proposition 1.2.8 at our disposal, this is immediate: the unique subgroup of E[p]
distinguished above is the first canonical subgroup of E. But more is needed to make possible the writing of the series: it is crucial that for any (E, H) inYan[0,0] all terms ofUp(E, H) apart from (E/G1,H) lie in a region which is admissibly disjoint from¯ Yan[0,0], which, in this case, is Yan[1,1]. This is the content of Equation 1.2.5 and is exactly what allows the decomposition of the Up correspondence as
Up =Upnsp+Upsp
on Yan[0,0]. This already gives us the first partial sum of the series, i.e., F1 = (1
ap)Unsp(f|Yan[1,1]), defined over S1:=Yan[0,0]. To write the second partial sum,
F2= ( 1
ap)Upnsp(f|Yan[1,1]) + (1
ap)2UpspUpnsp(f|Yan[1,1]),
we need to make sense of UpspUpnsp(f) which is formally defined on S2 := (Upsp)−1(S1) (which, again, happens to be Yan[0,0] in this case). Similarly, if we define, Sm := (Upsp)−m(S1), we can make sense of
Fm=
m
X
n=1
( 1 ap
)n(Upsp)n−1Upnsp(f|Yan[1,1])
as a section of ωk on Sm. In the case at hand, Sm happens to be Yan[0,0] for all m. But let us forget that knowledge and see what we can deduce about theSm’s, simply from their definition.
In fact, using Proposition 1.2.11, we can formally see that
Sm ={(E, H)∈Yan[0,1) :∃!(E/Gm,H)¯ ∈Upm{(E, H)} s.t. deg(E/Gm,H) = deg(E, H)}.¯ Let us call Sm thespecial locus of order m. Using Proposition 1.2.11, we formally deduce that
S1 ⊇ S2 ⊇ · · · ⊇ Sm⊇ · · ·
Therefore, the series given by the partial sums Fm can at least be written down on S∞:=∩m∈NSm,
provided it has a rigid analytic structure. The next step would be to show that the series converges onS∞given the slope condition. This boils down to estimating (a1
p)m(Upsp)m(f|Yan[1,1]) as in Lemma 1.2.16. This expression involves m iterations of Upsp, which, in turn, entails m applications of the pullback of differential forms under the map pr :E→E/G1 for various points (E, H)∈ S∞. By Lemma 1.2.3, an estimate can be obtained in terms of the degree of the various distinguished subgroups, i.e., the G1’s that appear in the iterations. In the case at hand, all the G1’s will be canonical subgroups of ordinary elliptic curves, and, hence, will be of degree 1, determining the slope condition νp(ap) < k−1 for the convergence of the series. In general, one expects these degrees to be large enough integers providing estimates which translate into relevant slope conditions.
Some issues remain to be handled. Firstly, if, unlike in the case at hand, S∞6=S1, we would still need to analytically continue f to S1. Secondly, we need to glue the section obtained via the above series to the section defined outside the special locus. Both of these require analytic continuation of Fm outside Sm. To do so, one needs to work in a strict neighborhood of the bad locusS1, say S0†, and construct the special locus of orderm inside S0†, calledSm†, which will certainly contain Sm. In fact, it would be enough to construct S1†, a neighborhood of S1 inside S0†, to have the following properties:
• the decompositionUp =Upnsp+Upsp overconverges from S1 toS1†,
• Up takes S0†− S1† to a region on whichf is already defined.
Having S1† at hand, one can define Sm† := (Upsp)−m+1(S1†), and construct Fm, the partial series of order m, on Sm†, as explained in the previous section. This shall explain the notation used in
§1.2. In fact, in that proof one hasSm =Yan[0,0], Sm† =Yan[0,ptm0 ].
In the Hilbert case, explained in §4, the above construction involves one extra step. Since at the m-th step of the argument, we glue Fm and f mod pm, we need to arrange for the domains of definition of Fm and f to form an admissible covering of S0†. The non-explicit nature of the argument does not allow us to rule out, for instance, the possibility thatS1†equalsS1. This would certainly cause trouble in the gluing procedure. To remedy this, one needs to enlarge Sm† (which already contains the “bad-behaviour” locus for Upm) to a strict neighborhoodSm†† ofSm†, in a way that the partial series Fm extends fromSm† toSm††. To arrange this, we essentially need to make sure that the above decomposition ofUp overconverges further yet, fromS1† toS1††. This step can be done using a general rigid analytic result on overconvergence of sections to finite ´etale maps between rigid analytic varieties ([Ber96, 1.3.5]).
Our hope is that the rather vague discussion in this section would serve as a “psychological”
preparation for the upcoming classicality arguments in the Hilbert case.
2. Hilbert Modular Varieties
In the upcoming sections, we intend to present two types of analytic continuation results for overconvergent Hilbert modular forms. The first will be results on “domains of automatic analytic continuation” for overconvergent Hilbert modular forms as in [Kas13, KST12], where no slope conditions are given (apart from the finiteness of slope). These results have been used in proving cases of the strong Artin conjecture in [Kas13, KST12]. The second type will be classicality results in the presence of slope conditions as in [PS11], where the method presented in§1 is used.
In preparation for the above, we will discuss the geometry of Hilbert modular varieties in this section, where the results are mostly from [GK12].
2.1. Notation. Let p be a prime number, L/Q a totally real field of degree g in which p is unramified, OL its ring of integers, dL the different ideal, and N an integer prime top. LetL+ denote the elements of L that are positive under every embedding L ,→R. For a prime ideal p of OL dividing p, let κp = OL/p, fp = deg(κp/Fp), f = lcm{fp :p|p}, and κ a finite field with pf elements. We identifyκp with a subfield ofκ once and for all. Let Qκ be the fraction field of W(κ). We fix embeddings Qκ ⊂Qurp ⊂Qp.
Let [Cl+(L)] be a complete set of representatives for the strict (narrow) class group Cl+(L) of L, chosen so that its elements are ideals a C OL, equipped with their natural positive cone a+=a∩L+. Let
B= Emb(L,Qκ) =`
pBp,
where p runs over prime ideals of OL dividing p, and Bp = {β ∈ B:β−1(pW(κ)) = p}. Let σ denote the Frobenius automorphism of Qκ, liftingx7→xp modulop. It acts on Bvia β7→σ◦β, and transitively on each Bp. For S⊆B we let
`(S) ={σ−1◦β:β ∈S}, r(S) ={σ◦β:β∈S}, and
Sc=B−S.
The decomposition
OL⊗ZW(κ) =M
β∈B
W(κ)β,
where W(κ)β is W(κ) with the OL-action given byβ, induces a decomposition, M =M
β∈B
Mβ,
on any OL⊗ZW(κ)-moduleM.
Let A be an abelian scheme over a scheme S, equipped with real multiplication ι: OL → EndS(A). Then the dual abelian scheme A∨ has a canonical real multiplication, and we let PA= HomOL(A, A∨)sym. It is a projective OL-module of rank 1 with a notion of positivity; the positive elements correspond to OL-equivariant polarizations.
For aW(κ)-scheme Swe shall denote by A/S, or simplyA if the context is clear, a quadruple:
A/S = (A/S, ι, λ, α),
comprising the following data: A is an abelian scheme of relative dimension g over a W(κ)- schemeS,ι:OL,→EndS(A) is a ring homomorphism. The map λis a polarization as in [DP94], namely, an isomorphism λ: (PA,PA+)→(a,a+) for a representative (a,a+)∈[Cl+(L)] such that A⊗OLa∼=A∨. The existence of λis equivalent, sincep is unramified, to Lie(A) being a locally free OL⊗ OS-module. Finally, α is a rigid Γ00(N)-level structure, that is,α:µN ⊗Zd−1L →A is an OL-equivariant closed immersion of group schemes.
Let X/W(κ) be the Hilbert modular scheme classifying such data A/S = (A/S, ι, λ, α). Let Y /W(κ) be the Hilbert modular scheme classifying (A/S, H), where A is as above and H is a finite flat isotropic OL-subgroup scheme ofA[p] of rank pg, where isotropic means relative to the µ-Weil pairing for someµ∈ PA+ of degree prime top. Let
π:Y →X
be the natural morphism, whose effect on points is (A, H)7→A.
Let X,X,Xrig be, respectively, the special fibre of X, the completion of X along X, and the rigid analytic space associated to Xin the sense of Raynaud. We use similar notationY ,Y,Yrig forY and letπ denote any of the induced morphisms. These spaces have models over Zp orQp, denoted XZp,Xrig,Qp, etc. For a point P ∈Xrig we denote byP = sp(P) its specialization in X, and similarly for Y. We denote the ordinary locus inX (respectively, Y) by Xord (respectively, Yord). Let Y0rig be the rigid analytic variety over W(κ) which classifies all (A, H, D) such that A∈Xrig,H, D are two subgroups ofA of the type classified by Yrig, and H∩D= 0. There are two morphisms π1, π2 :Y0rig→Yrig, whereπ1 forgetsD, and π2 quotients out by D.
2.2. The(ϕ, η)-invariant onY. LetQ∈Y correspond to (A, H) defined over a fieldk⊇κ. Let f:A→A/H be the natural projection andft:A/H →Abe the map induced by multiplication by p. We have ft◦f = [p]A and f ◦ft = [p]A/H. The natural maps induced by f, ft between the Lie algebras decompose as
M
β∈B
Lie(f)β: M
β∈B
Lie(A)β −→ M
β∈B
Lie(A/H)β, (2.2.1)
M
β∈B
Lie(ft)β: M
β∈B
Lie(A/H)β −→ M
β∈B
Lie(A)β. We define the following invariants of Qusing these maps:
ϕ(Q) =ϕ(A, H) ={β∈B: Lie(f)σ−1◦β = 0}, η(Q) =η(A, H) ={β∈B: Lie(ft)β = 0}, (2.2.2)
I(Q) =I(A, H) =`(ϕ(Q))∩η(Q) ={β∈B: Lie(f)β = Lie(ft)β = 0}.
The elements of I(Q) are thecritical indices of [Sta97]. By assumptionAsatisfies the Rapoport condition, and, hence, for any β ∈ B, both Lie(A)β and Lie(A/H)β are one-dimensional. Since
f ◦ft is multiplication by p = 0 on the Lie algebras, it follows that always at least one of the maps Lie(f)β and Lie(ft)β is zero for anyβ ∈B. This leads to the following definition.
Definition 2.2.1. A pair (ϕ, η) of subsets ofB is calledadmissible if `(ϕc)⊆η. Given another admissible pair (ϕ0, η0) we say that
(ϕ0, η0)≥(ϕ, η), if both inclusions ϕ0 ⊇ϕ, η0 ⊇η hold.
In the above definition, it is clear that if (ϕ, η) is admissible, then so is (ϕ0, η0), and that the admissibility of (ϕ, η) is equivalent tor(ηc)⊆ϕ. It is also easy to see that there are 3g admissible pairs.
Remark 2.2.2. IfH = Ker(FrA), thenϕ(A, H) =B. Similarly, IfH= Ker(VerA), thenη(A, H) = B. The invariant (ϕ, η) can be thought of as telling us for every direction β ∈ B whether H is Ker(FrA), Ker(VerA), or neither, even though the subgroup H does not necessarily decompose.
2.3. The type invariant on X. Let k be a perfect field of positive characteristic p. Let D denote the contravariant Dieudonn´e functor, G 7→ D(G), from finite commutative p-primary group schemesGoverk, to finite lengthW(k)-modulesM equipped with two maps Fr :M →M, and Ver : M →M, such that Fr(αm) = σ(α)Fr(m),Ver(σ(α)m) = αVer(m) for α ∈W(k), m∈ M and Fr◦Ver = Ver◦Fr = [p]. This functor is an anti-equivalence of categories and commutes with base change. It follows that ifGhas rankp` the length ofD(G) is`. Applying the Dieudonn´e functor to the Frobenius morphism FrG:G→G(p)givesD(FrG) :D(G(p))→D(G). This map, in view ofD(G(p)) =D(G)⊗W(k),σW(k), results in aσ-linear mapD(G)→D(G), which is nothing but the Frobenius morphism Fr of the Dieudonn´e module D(G). A similar statement is true for Ver. Let gi :G→Hi be morphisms, for i = 1,2. By considering g1×g2 :G→H1 ×H2, and applying the exactness ofD, it follows that
D(Ker(g1)∩Ker(g2)) =D(G)/(Im(D(g1) + Im(D(g2))).
(2.3.1)
Definition 2.3.1. Letkbe a perfect field of characteristicp. For an abelian schemeA/kclassified by X, thetype of A is a subset ofB defined by
(2.3.2) τ(A) ={β∈B:D(Ker(FrA)∩Ker(VerA))β 6= 0}.
If P is a point onX corresponding toA, we define τ(P) =τ(A).
2.4. The relationship between the type and the (ϕ, η) invariants.
Lemma 2.4.1. Let Q= (A, H) be a k-point ofY.
(1) β ∈τ(A) if and only if one of the following equivalent statements hold:
(a) Im(D(FrA))β = Im(D(VerA))β. (b) Im(Fr)β = Im(Ver)β.
(c) Ker(Fr)β = Ker(Ver)β.
(2) β ∈ϕ(A, H)⇐⇒Im(D(FrA))β = Im(D(f))β. (3) β ∈η(A, H)⇐⇒Im(D(VerA))β = Im(D(f))β.
Proof. Basic properties of Dieudonn´e modules recalled in§2.3 imply that
D(Ker(FrA)∩Ker(VerA)) =D(A[p])/(ImD(FrA) + ImD(VerA)) =D(A[p])/(Im Fr + Im Ver).
The modulesD(A[p]), Im(D(FrA)), and Im(D(VerA)) all have actions ofOL⊗ZW(κ), and, hence, decompose as a direct sum of their β-components in the usual way. Each D(A[p])β is a two- dimensionalk-vector space, and both Im(D(FrA))β and Im(D(VerA))β are one dimensional. This proves (1).
To prove (2) we recall the following commutative diagram:
0 //H0(A,Ω1A/k) //HdR1 (A/k) //H1(A,OA) //0
0 //D(Ker(FrA))⊗kk //D(A[p]) //D(Ker(VerA)) //0,
D(Ker(FrA(1/p)))
which is functorial in A. By duality, the map Lie(f) : Lie(A)→Lie(B) induces the map f∗: Lie(B)∗ =H0(B,Ω1B/k)→Lie(A)∗ =H0(A,Ω1A/k),
which is precisely the pull-back map f∗ on differentials. The mapf∗ has isotypic decomposition relative to the OL⊗k-module structure.
Now, β ∈ ϕ(f) ⇐⇒ Lie(f)σ−1◦β = 0 ⇐⇒ fσ∗−1◦β = 0. Via the identifications in the above diagram, the map f∗ can also be viewed as a map
f∗:D(Ker(FrB(1/p)))→D(Ker(FrA(1/p))), which is equal to the linear map D(f(1/p)|Ker(Fr
A(1/p))). So, fσ∗−1◦β = 0⇐⇒D(f(1/p)|Ker(Fr
A(1/p)))σ−1◦β = 0
⇐⇒D(f|Ker(Fr
A))β = 0.
We therefore have,
β∈ϕ(f)⇐⇒D(f|Ker(Fr
A))β = 0.
Now, D(f|Ker(FrA))β = 0 if and only if [D(Ker(FrA))/D(f)(D(Ker FrB))]β 6= 0 and that is equiva- lent to D(A[p])β/
D(f)(D(B[p])) +D(FrA)(D(A(p)[p]))
β 6= 0. By considering dimensions over k we see that this happens if and only if Im(D(f))β = Im(D(FrA))β, as the lemma states.
We first show that
Lie(ft)β = 0⇐⇒H1(f)β = 0.
Let γ ∈ PA be an isogeny of degree prime to p. In [GK12, Proof of Lemma 2.1.2], it is shown that there is an iγ∈ PB such that the following diagram is commutative:
(2.4.1) A γ //A∨
B
ft
OO
iγ //B∨.
f∨
OO
Applying Lie(·)β to the diagram, we obtain
Lie(A)β ∼= //Lie(A∨)β
Lie(B)β
Lie(ft)β
OO
∼= //Lie(B∨)β,
Lie(f∨)β
OO
and, hence,
Lie(ft)β = 0⇐⇒Lie(f∨)β = 0.
Since we have a commutative diagram:
Lie(A∨) ∼= //H1(A,OA)
Lie(B∨) ∼= //
Lie(f∨)
OO
H1(B,OB),
H1(f)
OO
where we can pass toβ-components, we conclude that Lie(ft)β = 0⇐⇒H1(f)β = 0.
The map H1(f) can be viewed as D(f|Ker(Ver
A)) :D(Ker(VerB))→D(Ker(VerA)), and hence, H1(f)β = 0 if and only if D(f|Ker(Ver
A))β = 0. This is equivalent to D(A[p])β/
h
D(f)(D(B[p])) +D(VerA)(D(A(p)[p])) i
β 6= 0.
Dimension considerations show that this happens if and only if Im(D(f))β = Im(D(VerA))β. We can now write down the relationship between the (ϕ, η) and τ.
Corollary 2.4.2. Let Q = (A, H) be a point of Y, and P = π(Q) = A a point of X. The following inclusions hold.
ϕ(Q)∩η(Q)⊆τ(P)⊆(ϕ(Q)∩η(Q))∪(ϕ(Q)c∩η(Q)c)
2.5. Definition of the strata. We define the stratum Wϕ,η on Y. We will show later that {Wϕ,η} is, indeed, a stratification ofY. First, we need a lemma.
Lemma 2.5.1. Given ϕ ⊆ B (respectively, η ⊆ B), there is a locally closed subset Uϕ, and a closed subset Uϕ+ (respectively, Vη and Vη+) of Y, such that Uϕ (respectively, Vη) consists of the closed pointsQ of Y withϕ(Q) =ϕ (respectively, η(Q) =η), and Uϕ+ (respectively, Vη+) consists of the closed point Q withϕ(Q)⊇ϕ (respectively, η(Q)⊇η).
